For (classical) primitive substitution tilings [1]}, the asymptotic growth of many quantities scales like \(\lambda ^n\) , where \(\lambda \)  is the associated inflation
factor — respectively its \(d\) -th power, if the tiling lives in \(\mathbb {R}^d\) , for \(d \ge 2\) . For example, the number of tiles of all types (or of any single tile type) in the \(n\) -th iteration of the substitution applied to one prototile grows like \(\Theta (\lambda ^n)\) . Both these phenomena originate from the uniqueness of the Perron–Frobenius eigenvalue \(\lambda \)  of the substitution matrix \(M\)  and from the Jordan form of \(M\)  and its powers \(M^n\) .
